
Alternating series are series whose terms alternate in sign between positive and negative. There is a powerful convergence test for alternating series.

Many of the series convergence tests that have been introduced so far are stated with the assumption that all terms in the series are nonnegative. Indeed, this condition is assumed in the Integral Test, Ratio Test, Root Test, Comparison Test and Limit Comparison Test. In this section, we study series whose terms are not assumed to be strictly positive. In particular, we are interested in series whose terms alternate between positive and negative (aptly named alternating series). It turns out that there is a powerful test for determining that a series of this form converges.

Compared to our convergence tests for series with strictly positive terms, this test is strikingly simple. Let us examine why it might be true by considering the partial sums of the alternating harmonic series. The first few partial sums with odd index are given by

Note that a general odd partial sum is of the form and the quantity in the parentheses is positive. We conclude that:

The sequence $\{s_1,s_3,s_5,\ldots \}$ of odd partial sums defined above is
increasing decreasing

Moreover, the sequence of odd partial sums is bounded below by zero, since and each quantity in parentheses is positive.

We conclude that the sequence $\{s_1,s_3,s_5,\ldots \}$ of odd partial must converge to a finite limit by applying
The Fundamental Theorem of Calculus. The Monotone Convergence Theorem. The Ratio Test.

Next consider the sequence

of even partial sums.

The sequence $\{s_2,s_4,s_6,\ldots \}$ of even partial sums defined above is
increasing and bounded, and therefore converges by the Monotone Convergence Theorem. decreasing and bounded, and therefore converges by the Monotone Convergence Theorem.

Finally, we use and the fact that the limits of the sequences $\{s_{2n+1}\}$ and $\{s_{2n}\}$ are finite to conclude that It follows that the alternating harmonic series converges.

With slight modification, the argument given above can be used to prove that the Alternating Series Test holds in general.

Does the alternating series test apply to the series
yes no